
Given \((t_k,q_k,p_k,E_k)\) , the coupled nonlinear equations (REF ) and () are solved implicitly to obtain \(q_{k+1}\)  and \(t_{k+1}\) . The configuration \(q_{k+1}\)  and time \(t_{k+1}\)  are then used in () and () to obtain \((p_{k+1},E_{k+1})\)  explicitly. The extended discrete Euler-Lagrange equations were first written in the time-marching form in [1]} and are also known as symplectic-energy-momentum integrators.
